In this paper, we introduce the notions of parallelizability and sections for semigroups. Such notion extend the concept of parallelizability already known for groups, where now we may not possess backward uniqueness of solutions. We study the relationship between this concept and the one of dispersiveness, and show that a dispersive semigroup with a compact section is parallelizable. We also present two applications. Firstly, for a semigroup <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_734_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\pi\)</EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_734_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^n\)</EquationSource> </InlineEquation> with a global attractor <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_734_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathscr {A}\)</EquationSource> </InlineEquation> satisfying the backward uniqueness property on a neighborhood of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_734_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathscr {A}\)</EquationSource> </InlineEquation>, we prove that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_734_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\pi\)</EquationSource> </InlineEquation> is parallelizable in <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_734_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^n\setminus \mathscr {A}\)</EquationSource> </InlineEquation>. This result can be replicated in infinite dimensional spaces with one additional assumption. Lastly, for a semigroup <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_734_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\pi\)</EquationSource> </InlineEquation> in a nonempty subset of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12591_2025_734_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^n\)</EquationSource> </InlineEquation> with a compact section with continuous function, we construct an impulsive semigroup with a global attractor.

Sections and Parallelizable Semigroups

期刊论文
发表日期：

2025年9月4日

查看全文

Everaldo M. Bonotto,
Matheus C. Bortolan,
Tiago A. Pacífico

摘要

In this paper, we introduce the notions of parallelizability and sections for semigroups. Such notion extend the concept of parallelizability already known for groups, where now we may not possess backward uniqueness of solutions. We study the relationship between this concept and the one of dispersiveness, and show that a dispersive semigroup with a compact section is parallelizable. We also present two applications. Firstly, for a semigroup \(\pi\) in \(\mathbb {R}^n\) with a global attractor \(\mathscr {A}\) satisfying the backward uniqueness property on a neighborhood of \(\mathscr {A}\) , we prove that \(\pi\) is parallelizable in \(\mathbb {R}^n\setminus \mathscr {A}\) . This result can be replicated in infinite dimensional spaces with one additional assumption. Lastly, for a semigroup \(\pi\) in a nonempty subset of \(\mathbb {R}^n\) with a compact section with continuous function, we construct an impulsive semigroup with a global attractor.